How to solve the simplest inequalities – one-step inequalities

One-step inequalities

One-step inequalities are pretty much self-explanatory – they are the type of inequalities that are solvable in a single step. Most of time, they’ll come in one of these forms:

Since we already learned what inequalities are, and we know the rules and symbols, we’ll concentrate on learning to solve one-step inequalities through several examples. So, let’s get started!

For example 1:

Find set for:

To solve any inequality, we need to “isolate the variable on one side”. In this case, we can do it in two ways. We can either subtract the number from both sides and then solve the expression on the right side.

Subtracting from both sides:

On the number line the solution is:

If we multiplying innequalities with negative number, inequality sign will be changed.

Now that we’ve calculated the result, we can present it in two ways: by writing it down as an interval and/or by marking it on the number line. For practice reasons, we’ll do it both ways.

So, this is how we would write down this result as an interval:

And this is how we would mark it on the number line:

Example 2:

Let’s try one with multiplication. How would we solve this problem?

As we can see, the only thing we needed to do was to multiply the whole inequality by the number . The solution of our inequality contains all numbers greater than number , as well as the number itself. This is due to the presence of the “greater or equal” sign in the inequality. In the form of an interval, the solution would be written down as:

And like this on the number line:

Example 3:

Let’s try one that requires division, but we’ll make it a bit more interesting. How would we solve this problem?

As we already said, a single division was required to solve this inequality, but this example required us to remember a very important information: when the variable changes signs, the inequality sign changes to its opposite as well! So, instead of a “greater than”, we end up with a “lesser than” at the end of our problem!

But all other things stay the same, so the solution looks like this in interval form:

And like this on the number line:

So, this is it for one-step inequalities. If you would like to practice some more, feel free to use the worksheets below.